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Can we say that every bounded continuous function is the limit of linear combination of continuous functions that vanish in infinity?In fact if X be a locally compact then can we say that ${\overline{lin(C_0(X))}}=C_b(X)$ or ${\overline{lin(C_0(X))}}=C(X)$ ?

mrf
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Hamid Shafie Asl
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1 Answers1

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First note that $C_b(X)$ is a linear subspace of $C(X)$, so we have $\def\lin{\mathop{\rm lin}}\lin C_b(X) = C_b(X)$. On the other hand we have $C_0(X) \subseteq C_b(X)$ as:

Let $f \in C_0(X)$, then there is a compact $K$ such that $|f|_{X \setminus K}|\le 1$, as $f$ is bounded on $K$ by compactness, $f$ is bounded on $X$, so $f \in C_b(X)$.

Regarding the question if $\overline{C_b(X)} = C(X)$ it depends on the topology we are looking at. As $C_b(X)$ is complete with respect to uniform convergence, we have $\overline{C_b(X)} = C_b(X)$ here, with respect to locally uniform convergence on locally compact $X$ or with respect to pointwise convergece, we have $\overline{C_b(X)} = C(X)$.

Sahiba Arora
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martini
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    It's better to change my question.We know $\ell^1(G)\subset M(G)\subset C_b(G)^$. I want to prove ${\overline{\ell^1(G)}}=M(G)$ respect to w-topology on $C_b(G)^*$. – Hamid Shafie Asl Sep 27 '13 at 09:14